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Control of Inverse Response Process using Model Predictive Controller
(Simulation)
by
Fawwaz bin Fuat
Dissertation submitted in partial fulfilment of
the requirements for the
Bachelor of Engineering (Hons)
(Chemical Engineering)
SEPTEMBER 2012
Universiti Teknologi PETRONAS
Bandar Seri Iskandar
31750 Tronoh
Perak Darul Ridzuan
I
CERTIFICATION OF ORIGINALITY
This is to certify that I am responsible for the work submitted in this project, that the
original work is my own except as specified in the references and acknowledgements,
and that the original work contained herein have not been undertaken or done by
unspecified sources or persons.
_____________________
FAWWAZ BIN FUAT
II
ABSTRACT
Model predictive control is an important model-based control strategy devised for large
multiple-input, multiple-output control problems with inequality constraints on the input
and outputs. Applications typically involve two types of calculations: (1) a steady-state
optimization to determine the optimum set points for the control calculations, and (2)
control calculations to determine the input changes that will drive the process to the set
points. The success of model-based control strategies such as MPC depends strongly on
the availability of a reasonably accurate process model. Consequently, model
development is the most critical step in applying MPC.
As Rawlings (2000) has noted, “feedback can overcome some effects of poor model, but
starting with a poor process model is a kind to driving a car at night without headlight.”
Finally the MPC design should be chosen carefully.
Model predictive control has had a major impact on industrial practice, with over 4500
applications worldwide. MPC has become the method of choice for difficult control
problems in the oil refining and petrochemical industries. However, it is not a panacea
for all difficult control problem(Shinkey, 1994; Hugo, 2000). Furthermore, MPC has had
much less impact in the order process industries. Performance monitoring of MPC
systems is an important topic of current research interest.
III
ACKNOWLEDGEMENT
Grateful……that’s the best word that I can say to appear my grateful after finished
completed final year project report at Universiti Teknologi PETRONAS. Firstly, I would
like to show my appreciation to my supervisor, Dr Lemma Dendena Tufa for supporting
me throughout this two semester and constantly provides me with assistance and help.
He is not only sacrificed their working time to supervise me to complete this project with
success but also taught me some moral virtues and how to behave a good employee in an
organization.
Secondly, I would like to thank the Chemical Engineering Department in
University of Technology PETRONAS (UTP) for giving me the opportunities to prepare
this technical report as my final year project. It’s has been help me a lot in refreshing
what I have learnt throughout five years I am here.
Furthermore, I would also like to express my thankfulness to my beloved parents
for giving me mentally supports to carry on this task.
Other than that, a special thanks to the reviewers of this report who contributed
significantly to project and to all of my friends, who offered valuable suggestion to
finished this report.
Finally, I would like to appear my hope that I can apply what I learnt when
finished my study soon.
IV
TABLE OF CONTENTS
CERTIFICATION . . . . . . . I
ABSTRACT . . . . . . . . II
ACKNOWLEDGEMENT . . . . . . III
CHAPTER 1: INTRODUCTION . . . . 1
1.1. Project Background . . . 1
1.2. Problem Statements . . . 1
1.3. Objectives and Scope of Study. . 2
CHAPTER 2: LITERATURE REVIEW . . . 3
CHAPTER 3: METHODOLOGY . . . . 16
3.1. Project Work . . 16
3.2 Research Methodology . . 18
3.3 Activities/Gantt Chart and Milestone . 19
CHAPTER 4: RESULT AND DISCUSSION . . 20
4.1 Data gathering and Analysis
4.2 Result and Discussion
CHAPTER 5: CONCLUSION AND RECOMMENDATION 45
5.1 Conclusion
5.2 Recommendation
REFERENCES . . . . . . . 46
APPENDICES
V
LIST OF FIGURES
Figure 1: Basic Structure of Model Predictive Controller
Figure 2: Model Predictive Controller block diagram
Figure 3: Project activities flow
Figure 4: MATLAB Simulation
Figure 5: Effect of Ƭi
Figure 6: Block diagram
Figure 7: Effect of Ƭi with Model Predictive Controller
Figure 8: Graph with Input rate weight Tuning
Figure 9: Graph with Output rate weight Tuning
Figure 10: Graph with Prediction Horizon Tuning
Figure 11: Graph with Control Horizon Tuning
Figure 12: Feedback Control Loop
Figure 13: Servo Control
Figure 14: Graphical Interpretation of IAE
Figure 15: Input (bottom) and output (top) data used for identification
Figure 16: The Bode plot of the actual model
Figure 17: Bode plot of the identified model and the actual model
VI
LIST OF TABLE
Table 1: Ghantt Chart
Table 2: Error with different Ƭ푖
Table 3: Error with default MPC setting with Ƭi Tuning
Table 4: Error with Input rate weight Tuning
Table 5: Error with Output rate weight Tuning
Table 6: Error with Prediction Horizon Tuning
Table 7: Error with Control Horizon Tuning
1
CHAPTER 1
INTRODUCTION
1.1 Project Background
Systems which have inverse responses are difficult to control. Many types of process
show inverse response over a part of the normal range operation among these is the
recycling system so common in engineering practice. Control can be greatly improved
by introducing a compensating feedback element into the control loop. In this project,
we will use Model Predictive Controller to control the inverse response processes.
1.2 Problem Statement
Some effect to the behavior of the process is affected by the tuning of the Model
Predictive Controller. The time taken to reach the set point and the error may be affected
by the parameter tuning for inverse response process. Based on the system, we have to
measure the error of the controller and make a comparison of efficiency between
response with and without using controller.
1.3 Objectives and Scope of Study
I. Analyze the effect of Ƭi response without using controller for a typical inverse
response process by MATLAB software.
II. Analyze the effect of Ƭi response by using Model Predictive Controller for a
typical inverse response process by MATLAB software.
III. Analyze the effect of input rate weight response by using Model Predictive
Controller for a typical inverse response process by MATLAB software.
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IV. Analyze the effect of output rate weight response by using Model Predictive
Controller for a typical inverse response process by MATLAB software.
V. Analyze the effect of prediction horizon response by using Model Predictive
Controller for a typical inverse response process by MATLAB software.
VI. Analyze the effect of control horizon response by using Model Predictive
Controller for a typical inverse response process by MATLAB software.
In order to study process control of inverse responses, the transfer functions should first
be developed. The transfer function which is the ratio of the output variable X to the
input signal Y in the Laplace Transformations is defines the performance of the process.
These equations represent a variety of typical combined processes or their
approximations. For this project, we will use second order equation of transfer function.
3
CHAPTER 2
LITERATURE REVIEW
The analysis, design, control, operation, and optimization of distillation columns
was studied and researched for long time ago [1,2]. As time goes by, engineers try to
come out with techniques and approaches to stabilize the control process due to the
present of nonlinearities and constraints. The most famous and popular method of model
predictive control is using the Dynamic Matrix Control (DMC) [3,4]. It is based on a
discrete time step response model that calculates a desired value of manipulated value
that remains unchanged during the next time step.
One of the applications of DMC is in multiple input and multiple output systems
(MIMO). In simulation experiment, MIMO is always compared in term of performance
with Single Input and Single Output (SISO) process system and the standard of model
predictive control. By this performance measurement, we can know which controller is
the most suitable for certain control process.
Besides, DMC introduced by Culter and Ramaker is available in most
commercial industrial distributed control systems and process simulation software
packages [5]. The three integral parts of any model predictive control algorithm which
are process model, the cost function and the optimization technique. There are various
forms of MPC such as Robust MPC that provides guaranteed feasibility and stability of
the process, feedback MPC that mitigates shrinkage of feasible region, pre-computed
MPC that using linear or quadratic programming, and decentralized MPC that can give
fast response.
4
Basic structure of MPC
The major contributions of MPC in control system industry are it handles
structural changes, easy to tune method, allow operation closer to constraint that can
increase the profit, can take account of actuator limitations, has plenty of time for on-line
computations, can handle non-minimal phase and unstable processes, and can handle
multivariable control problems naturally. The applications of MPC are always used in
distillation column, hydrocracker, pulp and paper plant, servo mechanism and robot arm.
The OBF models can be considered as a generalization of Finite Impulse
Response (FIR) models in which the filter are replaced with more complex orthonormal
basis filters [6,7]. There are many types of OBF such as Laguerre filter which has one
real pole for damped processes, Kautz filter which allows the incorporation of a pair
conjugate complex pole for modeling weakly damped processes, Generalized
orthonormal basis filter and Markov OBF which for a system involves time delay and
time delay estimation.
In many control application the desired performance cannot be expressed caused
by the process constraint [8,9]. Example for physical constraint is the actuator limits,
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safety constraint such as temperature and pressure limits, performance constraint such as
overshoot and others constraints like manipulated variable constraint which hard limits
on input, manipulated variable rate constraint which are hard limit on the size of the
manipulated variable, and output variable constraint which hard or soft limits on the
output on the system are imposed to.
Based on my research, the advantages of using MPC are it is a straight forward
formulation based on well understood concepts, it can handle constraints, the
development time much shorter than for competing advanced control methods, easier to
maintain the changing model or the specs does not require complete redesign [10]. By
running different scenarios in linear and nonlinear simulations, you can evaluate
controller performance. You can adjust controller performance as it runs by tuning
weights and varying constraints [11].
In addition to developing more flexible control technology, new process
identification technology was developed to allow quick estimation of empirical dynamic
models from test data, substantially reducing the cost of model development [12]. More
importantly, MPC theory can lead to discoveries by which MPC can be improved
through new formulation [13]. This is important to make sure the efficiency of the
control process in minimizing the constraint and improve the control system in industry
or any manufacturing.
I. Model Predictive Control
Introduction to MPC
A general development is presented that gives great insight into the roles of both the
control algorithm and the process in the behavior of feedback systems. It also provides a
method for tailoring the feedback control algorithm to each specific application. Since
the model of the process is an integral part of the control algorithm, the controller
equation structure depends on the process model. Although the control algorithm is
different, the feedback concept is unchanged, and the selection criteria for manipulated
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and controlled variables are the same. The algorithms could be used as replacements for
the PID controller in nearly all applications. PID controller algorithm is considered the
standard algorithm. Alternative algorithm is selected only when it provides better
performance.
MPC Structure
Consider the typical thought process used by a human operator implementing a feedback
control manually. The approach used by the operator has three important characteristics:
I. It uses a model of the process to determine the proper adjustment to the
manipulated variable , because the future behavior of the controlled variable can
be predicted from the values of the manipulated variable.
II. The important feedback information is the difference between the predicted
model response and the actual process response. If this difference is zero, the
control would be perfect, and no further correction would be needed
III. This feedback approach can result in the controlled variable approaching its set
point after several iterations, even with modest model errors
A continuous version of the approach can be automated with the general predictive
control structure.
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The variable Emis equal to the effect of the disturbance, Gd(s)D(s), if the model is perfect
(Gm(s)=Gp(s)). The structure highlights the disturbance for feedback correction.
However, the model is essentially never exact. The feedback signal includes the effect of
the disturbance and the model error, or mismatch. The feedback signal can be considered
as a model correction. It is used to correct the set point so as to provide a better target
value, Tp(s), to the predictive control algorithm. The controller calculates the value of the
manipulated variable based on the corrected target.
The closed-loop transfer functions for the setpoint and disturbances are:
The controller algorithm, Gcp(s), for the predictive structure is to be determined to give
good dynamic performance. Let us determine a few properties of the predictive structure
that establish important general features of its performance and give guidance for
designing the controller. A very important control performance objective is to ensure
that the controlled variable returns to its set point in steady state. This objective can be
evaluated from the closed-loop transfer functions by applying the final value theorem
)()()(1)()(
)()()()()(1)()()(
)(SP)(CV
sGsGsGsGsG
sGsGsGsGsGsGsGsG
ss
mpcp
pcp
mspvcp
pvcp
)()()(1
)()()(1)()()()()(1
)()()(1)()(CV
sGsGsGsGsGsG
sGsGsGsGsGsGsGsG
sDs
mpcp
dmcp
mspvcp
dmcp
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and determining whether the final value of the controlled variable, expressed as a
deviation variable from the initial set point, reaches the set point.
The next control performance objective is perfect control. Perfect control means the
controlled variable never deviates from the set point.
CV(s)/D(s) = 0 and CV(s)/SP(s) = 1 provide the basis for the following condition:
Perfect control can be achieved if the controller could be set equal to the inverse of the
process dynamic model. Block diagram algebra can be applied to derive the following
condition for the behavior of the manipulated variable.
The perfect control system must invert the true process in some manner.
)()( 1 sGsG mcp
)()(
1)()(1)()(
)()()(1)()(
)()(MV
sGsG
sGsGsGsG
sGsGsGsGsG
sDs
p
d
pcp
cpd
mpcp
cpd
9
II. Inverse Response
Inverse response behavior appears when the initial response of the output variable is in
the opposite direction to the steady state value. In chemical process industry this
phenomenon occurs in several systems such as distillation column and drum boilers [14].
The reason for the inverse response is that the process transfer function has an odd
number of zeros in the open right half plane. This non-minimum phase characteristics of
the process affects the achievable closed-loop performance because the controller
operates on wrong sign information in the initial time of the transient [15]. This fact
introduces essential limitations in terms of achievable output performance.
Let assume a process resulting from two parallel first-order stable processes having
opposite gain
Where K1, K2, τ1, and τ2 are positive constants. The overall transfer function of P(s) in
equation above can be posed as
Where
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And
In this context, inverse response appears due to competing effects of slow and fast
dynamics. In concrete terms, it appears when the slower process has higher gain.
Therefore, the condition for inverse response reads as:
Study of Other Controller
Dynamic Matrix Control (DMC)
DMC is one of the most popular methods of model predictive control. A way to have
students explore the nature of DMC control is to use it on a simulated process [16].
DMC control is based on a discrete time step response model that calculates a desired
value of the manipulated value that remains unchanged during the next time step. The
new value of the manipulated variable the value that gives the smallest sum of squares
error between the set point and the predicted value predicted values of the controlled
variable. The dynamic model used to predict the future values of the controlled variable
is represented by a vector A, whose elements are defined as
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Whrere
y(t) = the value of the controlled variable at time t
∆y(t0) = the change in the manipulated variable at t0
Thus, the response of a process to a step change ∆u, in the manipulated variable at t0
∆u(t0) is given by
Orthonormal Basis Control (OBF)
General orthonormal basis filter (OBF) models are generalization of FIR models,
presented a least square identification method to estimate a finite number of model
parameters based on Pulse, Laguerre and Kautz basis functions. They showed that the
form of the chosen basis functions and the accuracy of the poles used in them can
improve the identification results.
A test was simulated on the transfer function given and an output data was generated for
a given input data sequence with additive white noise of power 2.
12
An OBF model was developed using the input-output data with an initial time constant
τ=3, and time delay=0. It is observed that the resulting OBF model with only 5 OBF
terms has a very good accuracy.
Figure 15: Input (bottom) and output (top) data used for identification
The Bode plot of the actual model and that which is identified by the discussed
technique is presented below. As it can be seen, through there is small difference in the
time constants, and gain however, the Bode plots are nearly indistinguishable, hence
good enough for controller design purpose.
13
Figure 16: The Bode plot of the actual model
The response of the OBF model and the noisy output data plotted together for the same
input
Figure 17: The Bode plot of the identified model and the actual model
Internal Method Controller (IMC)
IMC controller is based on Brosilow (1979) and Garcia and Morari (1983). Since an
exact inverse is not possible, the IMC approach segregates and eliminates the aspects of
the model transfer function that make the calculation of realizable inverse impossible.
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Thus, the first step to start the IMC approach is to factor the model into the product of
two facors.
The noninvertible part has an inverse that is not causal or is unstable
The steady state gain of this term must be 1.0
The invertible part has an inverse that is causal and stable, leading to
realizable, stable controller
The IMC Controller (idealized)
This design ensures the controller is realizable and the system is internally stable
Example of IMC Controller
To make the controller proper or semiproper, add a filter to make the controller
proper
For tracking setpoint changes,
Adjust the filter-tuning parameter to vary speed of the response of the closed-
loop system
)()()( sGsGsG mmm
)(sGm
)(sGm
1)()(~ sGsG mcp
3)15(039.0)(
s
sGm
3)15(039.0)(
ssGm 0.1)( sGm
039.0
)15()()(3
1
ssGsG mcp
)()()(~)(1 sGsGsGsG fmcpcp
nf ssG
11)(
15
: small --- response is fast
large --- the closed loop response is more robust (insensitive to model error)
Example:
The controller is proportional-derivative, which still might be too aggressive but can be
modified to give acceptable performance. Then, to make the controller semiproper,
115.10
039.01)()(
s
ssGsG cpf
)15.10(039.0)(
5.5
sesG
s
m )15.10(039.0)(
ssGm
sm esG 5.5)(
039.0
)15.10()()(~ 1
ssGsG mcp
1039.0)15.10()(
sssGcp
16
CHAPTER 3
METHODOLOGY/PROJECT WORK
3.1 Project work
Start
Research
Literature Review
Choosing Model Predictive Controller
Simulation
Result and Discussion
Final report
End
Figure 3: Project Activities Flow
Figure 4: MATLAB Simulation
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The project is a simulation project. Specifically, it is a simulation of Model Predictive
Controller on how the performance to system. First and for most, the project will begin
with the research on several issues which had been mention in the research methodology
above.
With the collective information, the project will proceed with the literature review that
will mention about what people had done before for Model Predictive Controller
strategies and the performance based on certain process. Besides, the author will discuss
a basic knowledge of on how the Model Predictive Controller functioning and what are
the benefits when using Model Predictive Controller in a control system or process.
After completing the literature review, the further studies will move on to tune the
Model Predictive parameter of Controller approaches to make a performance
comparison between them due to a process stated in the objective of the proposal.
Besides, the author needs to identify the parameters that involved in measuring the
performance of each Model Predictive Controller due to system tested such as the value
Ƭi, input rate weight, output rate weight, prediction horizon and control horizon.
Then, the simulation of Model Predictive Controller will be done by using MATLAB.
After completing the simulation, the result and discussion will be done to know the
performance of each Model Predictive Controller based on the stated systems.
Lastly, all the studies and discussion will be compiled in the final report. Apart from
that, the justification of performance of the Model Predictive Controller also will be
mentioned in the discussion.
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3.2 Research Methodology
Research is a method taken in order to gain information regarding the major scope of the
project. The sources of the research cover the handbook of condensate stabilization unit,
e-journal, e-thesis and several trusted link.
The steps of research:
1. Gain information of the Model Predictive Controller strategies and its
performance.
2. List down the parameter used to be tested for inverse response process.
3. Make a comparison between the parameter tuned based on a control process
decided.
Project Simulation
The project simulation will be done by using MATLAB. Then, we will make a
comparison between parameter tuned based on a system that we already choose early in
the topic.
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3.3 Activities/Gantt Chart and Milestone
No Detail/Week 1 2 3 4 5 6 7
8 9 10 11 12 13 14 15
1 Project Work Continues
2 Submission of Progress Report
3 Project Work Continues
4 Pre- EDX
5 Submission of Draft Report
6 Submission of Dissertation (soft bound)
7 Submission of Technical Paper
8 Oral Presentation
9 Submission of Dissertation (hard bound)
20
CHAPTER 4
RESULT AND DISCUSSION
THE EFFECT OF Ƭi WITHOUT CONTROLLER
Ƭi = 2 Ƭi = 4
Ƭi = 6 Ƭi = 8
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Comparison between Ƭi = 2, 4, 6, and 8
Figure 5: Effect of Ƭi
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Overall process for Ƭi = 2, 4, 6, and 8
Figure 6: Block Diagram
Error for Ƭi = 2, 4, 6 and 8
CV Ƭi
Input variation
(Error)
Output Variation
(Error)
CV1 2 1.0672 15.982
CV2 4 1.1896 17.9701
CV3 6 1.374 19.9583
CV4 8 1.5585 21.9464
Table 2: Error of different Ƭi
23
DISCUSSION FOR THE EFFECT OF Ƭi WITHOUT CONTROLLER
For the inverse response of second order process with different value of Ƭi, we can see
the behavior of each graph changed with step value of 1. When we increase the value of
Ƭi, the inverse response become lower compared to smaller value of Ƭi. By using
MATLAB software, we calculate the error without using any controller in the process
and make comparison between them. Based on the graph and data calculated, we can say
that lower value of Ƭi is better and more efficiency to the process because it gives lower
value of error for the system although without using any model controller.
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THE EFFECT OF Ƭi WITH DEFAULT MPC SETTING
Ƭi = 2
Ƭi = 4
25
Ƭi = 6
Ƭi = 8
Figure 7: Effect of Ƭi with MPC
26
DATA AND ERROR VALUE FOR THE EFFECT OF Ƭi WITH DEFAULT MPC
SETTING
Table 3: Error with default MPC setting with Ƭi tuning
ti=2 ti=4 ti=6 ti=8
5.6 0 0 0
5.52 0.08 5.6 5.6 1.4 1.4 1.4 1.4
3.69 1.83 6.34 0.74 0.998 0.402 1.05 0.35
1.95 1.74 5.47 0.87 0.646 0.352 0.722 0.328
0.875 1.075 4.14 1.33 0.558 0.088 0.628 0.094
0.422 0.453 2.72 1.42 0.579 0.021 0.634 0.006
0.371 0.051 1.43 1.29 0.612 0.033 0.657 0.023
0.505 0.134 0.371 1.059 0.634 0.022 0.676 0.019
0.679 0.174 -0.366 0.737 0.651 0.017 0.706 0.03
Total 5.537 -0.762 0.396 0.665 0.014 0.692 0.014
Total 13.442 Total 2.349 Total 2.264
IAE 6.5922 IAE 9.3456 IAE 10.4738 IAE 11.1559
ISE 6.125 ISE 11.4942 ISE 11.0621 ISE 12.5704
ITAE 21.5067 ITAE 35.3649 ITAE 50.6487 ITAE 54.489
27
DISCUSSION FOR THE EFFECT OF Ƭi WITH DEFAULT MPC SETTING
For this controller tuning, we set all parameter in default setting such as input rate
weight=0.1, output rate weight=1.0, prediction horizon=10 and control horizon=2. We
used Ƭi=2, 4, 6 and 8 for this tuning to see the behavior of the graph after it been
simulated and make comparison between the process without using any controller. After
the simulation was run, the error value of Ƭi=2 using MPC controller is lower that is
6.5922 compare to the process without controller which is 15.982. The different between
these two errors is 9.3898. it is same goes to Ƭi=4, 6, and 8. The error becomes lower
because the MPC can predict the past input or output to become a predicted output and
come out with future error. Then, the future error undergoes optimization process to
reduce the cost function and constraint and finally come out with future input which will
send back to the controller. Then, the MPC will send the corrective signal to the set point
to minimize the error. Thus, for second order process it is better to use small number of
Ƭi in the transfer function.
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EFFECT OF MPC INPUT RATE WEIGHT= 0.1, 1, 10 TUNING FOR Ƭi = 6, OTHERS
ARE DEFAULT.
Rate Weight=0.1
Rate Weight=1.0
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Rate Weight=10
Figure 8: Graph with Input rate Weight Tuning
30
Table 4: Error with Input Rate Weight Tuning
weight=0.1 weight=1 weight=10
0 0 0
1.4 1.4 -0.471 0.471 -0.00639 0.00639
0.998 0.402 -0.876 0.405 -0.0128 0.00641
0.646 0.352 -1.23 0.354 -0.0191 0.0063
0.558 0.088 -1.54 0.31 -0.0255 0.0064
0.579 0.021 -1.84 0.3 -0.0319 0.0064
0.612 0.033 -2.13 0.29 -0.0382 0.0063
0.634 0.022 -2.42 0.29 -0.0446 0.0064
0.651 0.017 -2.72 0.3 -0.0509 0.0063
0.665 0.014 -3.05 0.33 -0.0573 0.0064
Total 2.349 -3.41 0.36 -0.0636 0.0063
Total 3.41 Total 0.0636
IAE 10.4738 IAE 8.092 IAE 9.9656
ISE 11.0621 ISE 6.6703 ISE 9.9314
ITAE 50.6487 ITAE 37.6566 ITAE 49.7679
31
DISCUSSION FOR THE EFFECT OF MPC INPUT RATE WEIGHT= 0.1, 1, 10
TUNING FOR Ƭi = 6, OTHERS ARE DEFAULT.
For this tuning of input rate weight, we only use Ƭi=6 with different value of input rate
weight which are 0.1, 1.0 and 10. The default value of input rate weight is 0.1. Based on
the graph above, the increment of input rate weight value from 0.1 to 1.0 decreases the
error value for the controlled variable and decrease the move sizes significantly,
especially the initial move. The controller also minimizes the weighted sum of
manipulated variable deviations from their nominal values. But for manipulated variable,
the error is increase a bit only when the input rate weight is increase, then it decreases to
0.0636 when we increase the weight rate to 10. The value of input rate weight is affected
to both manipulated and controlled variable error. So it is important to tune the input rate
weight to ensure that the error is the least for manipulated and controlled variables
respectively.
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EFFECT OF MPC OUTPUT RATE WEIGHT= 0.1, 1, 10 TUNING FOR Ƭi = 6,
OTHERS ARE DEFAULT.
Rate Weight=0.1
Rate Weight=1.0
33
Rate Weight=10
Figure 9: Graph with Output Rate Weight Tuning
output rate weight
weight=0.1 weight=1 weight=10
0 0 0
-0.471 0.471 1.4 1.4 3.47 3.47
-0.876 0.405 0.998 0.402 0.867 2.603
-1.23 0.354 0.646 0.352 0.852 0.015
-1.54 0.31 0.558 0.088 0.868 0.016
-1.84 0.3 0.579 0.021 0.878 0.01
-2.13 0.29 0.612 0.033 0.887 0.009
-2.42 0.29 0.634 0.022 0.895 0.008
-2.72 0.3 0.651 0.017 0.903 0.008
-3.05 0.33 0.665 0.014 0.91 0.007
-3.41 0.36 Total 2.349 Total 6.146
Total 3.41
34
Table 5: Error with Output Rate Tuning
DISCUSSION FOR THE EFFECT OF MPC OUTPUT RATE WEIGHT= 0.1, 1, 10
TUNING FOR Ƭi = 6, OTHERS ARE DEFAULT.
For this tuning of input rate weight, we only use Ƭi=6 with different value of output rate
weight which are 0.1, 1.0 and 10. The default value of output rate weight is 1. The
output weights let you dictate the accuracy with which each output must track its
setpoint. Specifically, the controller predicts deviations for each output over the
prediction horizon which we set it as default, 10. It multiplies each deviation by the
output's weight value. The weights must be zero or positive. A large weight on a
particular output causes the controller to minimize deviations in that output. Based on
the graph above, the increment of output rate weight value from 0.1 to 1.0 decreases the
error value for the controlled variable and decrease the move sizes significantly,
especially the initial move. The controller also minimizes the weighted sum of
manipulated variable deviations from their nominal values. But for manipulated variable,
the error is decrease a bit only when the input rate weight is increase, then it increases to
6.146 when we increase the weight rate to 10. The value of output rate weight is affected
to both manipulated and controlled variable error. So it is important to tune the input rate
weight to ensure that the error is the least for manipulated and controlled variables
respectively.
IAE 8.092 IAE 10.4738 IAE 10.5936
ISE 6.6703 ISE 11.0621 ISE 11.5651
ITAE 37.6566 ITAE 50.6487 ITAE 49.2717
35
EFFECT OF MPC PREDICTION HORIZON= 2, 10, 50 TUNING FOR Ƭi = 6,
OTHERS ARE DEFAULT.
Prediction Horizon= 2
Prediction Horizon= 10
36
Prediction Horizon= 50
PH=2 PH=10 PH=50
0 0 0
-5.25 5.25 1.4 1.4 5.95 5.95
-5.58 0.33 0.998 0.402 2.93 3.02
-5.48 0.1 0.646 0.352 1.25 1.68
-6.07 0.59 0.558 0.088 0.982 0.268
-7.25 1.18 0.579 0.021 1.05 0.068
-8.8 1.55 0.612 0.033 1.09 0.04
-10.6 1.8 0.634 0.022 1.08 0.01
-12.8 2.2 0.651 0.017 1.07 0.01
-15.3 2.5 0.665 0.014 1.06 0.01
-18.3 3 Total 2.349 Total 11.056
Total 18.5
Figure 10: Graph with Prediction Horizon Tuning
37
Table 6: Error with Prediction Horizon Tuning
DISCUSSION FOR THE EFFECT OF MPC PREDICTION HORIZON= 2, 10, 50
TUNING FOR Ƭi = 6, OTHERS ARE DEFAULT.
Prediction horizon is the number of control intervals over which the outputs are to be
optimized. Based on the data calculated, the error for the changes of prediction horizon
is not in sequence, means it is not increasing or decreasing based on the prediction
horizon number. When we change the prediction horizon from 2 to 10, the input error is
decreasing too much but the output error is increasing from 2.0298 to 10.4738. After
that when the prediction horizon is increase to 50, the input error is increasing and the
output error is increasing a bit only. So it is important to tune the prediction horizon
number to ensure that the error is the least for manipulated and controlled variables
respectively. Different tuning will give different behavior of input and output graph
which may affect the efficiency and time taken for the process to reach the set point.
IAE 2.0298 IAE 10.4738 IAE 10.6223
ISE 1.6624 ISE 11.0621 ISE 12.8529
ITAE 2.331 ITAE 50.6487 ITAE 44.7367
38
EFFECT OF MPC CONTROL HORIZON= 1, 5, 10 TUNING FOR Ƭi = 6, OTHERS
ARE DEFAULT.
Control Horizon= 1
Control Horizon= 2
39
Control Horizon= 10
Figure 11: Graph with Control Horizon Tuning
CH=1 CH=5 CH=10
0 0 0
-3.54 3.54 -3.29 3.29 -5 5
-3.65 0.11 -3.27 0.02 -5.27 0.27
-3.94 0.29 -2.97 0.3 -5.16 0.11
-4.48 0.54 -3.06 0.09 -5.68 0.52
-5.23 0.75 -3.45 0.39 -6.73 1.05
-6.15 0.92 -3.98 0.53 -8.12 1.39
-7.25 1.1 -4.58 0.6 -9.75 1.63
-8.53 1.28 -5.24 0.66 -11.7 1.95
-10 1.47 -5.96 0.72 -13.9 2.2
-11.7 1.7 -6.77 0.81 -16.5 2.6
Total 11.7 Total 7.41 Total 16.72
40
Table 7: Error with Control Horizon Tuning
DISCUSSION FOR THE EFFECT OF MPC CONTROL HORIZON= 1, 5, 10 TUNING
FOR Ƭi = 6, OTHERS ARE DEFAULT.
Control horizon sets the number of control intervals over which the manipulated
variables are to be optimized. Based on the data calculated, the error for the changes of
control horizon is not in sequence, means it is not increasing or decreasing based on the
control horizon number inserted. When we change the control horizon from 1 to 5, the
input error is decreasing too much but the output error is increasing from 4.3855 to
6.4548. After that when the control horizon is increase to 10, the input error is
increasing and the output error is decreasing to 2.5361. So it is important to tune the
control horizon number to ensure that the error is the least for manipulated and
controlled variables respectively. Different tuning will give different behavior of input
and output graph which may affect the efficiency and time taken for the process to reach
the set point.
IAE 4.3855 IAE 6.4548 IAE 2.5361
ISE 2.5723 ISE 4.5287 ISE 1.7408
ITAE 16.7406 ITAE 31.5615 ITAE 6.0361
41
Feedback Controller.
Feed forward controllers also exist but are more complicated to implement. Here we will
describe the use of feedback controllers.
The purpose of a servomechanism is to provide one or more of the following objectives:
I. accurate control of motion without the need for human attendants (automatic
control)
II. maintenance of accuracy with mechanical load variations, changes in the
environment, power supply fluctuations, and aging and deterioration of
components (regulation and self-calibration)
III. control of a high-power load from a low-power command signal (power
amplification)
IV. control of an output from a remotely located input, without the use of mechanical
linkages (remote control, shaft repeater).
In feedback control the variable required to be controlled is measured. This measurement
is compared with a given set-point. The controller takes this error and decides what
action should be taken by the manipulated variable to compensate for and hence remove
the error.
Figure 12: Feedback Control Loop
42
The advantage of this type of control is that it is simple to implement. Not only does the
feedback control system require no knowledge of the source or nature of the
disturbances, but it also requires minimal detailed information about how the process
itself works. Feedback control action is entirely empirical. So long as an adjustment is
being made in the correct sense then the control system should remove the effect of an
external disturbance.
A servo control loop is one which responds to a change in setpoint. The setpoint may be
changed as a function of time (typical of this are batch processes), and therefore the
controlled variable must follow the setpoint.
Figure 13: Servo Control
Response of second order process
Second order transfer function can arise physically whenever two first-order processes
are connected in series. For example, two stirred-tank blending processes, each with a
first-order transfer function relating inlet to outlet mass fraction, might be physically
connected so the outflow stream of the first tank is used as the inflow stream of the
second tank. Figure below illustrates the signal flow relation for such a process. Here
(ݏ)ܩ = (ݏ)(ݏ) =
2ܭ1ܭݏ) + ݏ2)(1 + 1) =
ܭݏ) + ݏ2)(1 + 1)
43
Calculating the Error
Controller tuning have been developed that optimize the response for a simple process
model and set point change. The optimum settings minimize an integral error criterion.
Three popular integral error criteria are:
1. Integral of the absolute value of the error (IAE)
ܧܣܫ = න ݐ|(ݐ)|∞
Where the error signal e(t) is the difference between the set point and the
measurement.
2. Integral of the squared error (ISE)
ܧܫ = න ݐଶ(ݐ)∞
3. Integral of the time-weighted absolute error (ITAE)
ܧܣܫ = න |(ݐ)|ݐ∞
ݐ
ܭ1 + 1
ܭ
2 + 1
X(s) U(s) Y(s)
Table 14: Graphical interpretation of IAE.
44
The ISE criterion penalizes large errors, while the ITAE criterion penalizes errors that
persist for long periods of time. In general, the ITAE is the preferred criterion because it
usually results in the most conservative controller settings. By contrast, the ISE criterion
provides the most aggressive settings, while the IAE criterion tends to produce controller
settings that are between those for the ITAE and ISE criterion.
45
CHAPTER 5
CONCLUSION
In conclusion, this project is to analyze the behavior of Model Predictive controller for
parameter tuning due to a system of inverse response control process. As the result of the
behavior measurement, we should have a basic knowledge on how to tuning the
controller for inverse response process. This is because inverse response is difficult to
control and the tuning may affect either the input or output error measurement. Besides,
by using Model Predictive controller is better than that not using any controller for any
process to minimize the error and increase the efficiency of the process.
46
REFERENCES
[1] HesamKomariAlaei, KarimSalahshoor, HamedKomariAlaei, 2010, Model Predictive Control of Distillation Column bases Recursive Parameter Estimation Method Using HYSYS Simulation. Page 308.
[2] Adrian Wills, March 2003, Doctor of Philosophy, Barrier Function Based Model Predictive Control.
[3] Charles R. Nippert, Simple Models That Illustrate Dynamic Matrix Control
[4] Sachin C. Patwardhan, Workshop on Model Predictive Control.
[5] RashmiRanjankar, AngshumanKalita, Dynamic Matrix Control
[6] D.T. Lemma, M. Ramasamy and M. Shuhaimi, 2010, System Identification using Orthonormal Basis Filters.Pages 2517.
[7] JieZeng, Raymond de Callafon, November 5-11, 2005, Filtersparametrized by Orthonormal Basis Functions for active noise control. Pages 1.
[8] Jan Maciejowski, Introduction to Predictive Control,
[9] Manfred Morari, Jay E. Lee, and Carlos E. Garcia, March 15, 2002, Model Predictive Control.pages 35.
[10] Glorinevsky, 2005, Lecture Note, The Concept of Model Predictive Control
[11] Model Predictive Control Toolbox, <http://www.mathworks.com/products/mpc/.
[12] Jan Maciejowski, Introduction to Predictive Control,
[13] Glorinevsky, 2005, Lecture Note, The Concept of Model Predictive Control
[14] F.G. Shinskey. Process control system.Mcgraw-Hill, 2nd. Edition,1979.
[15] S.Alcantara, C. Pedret, R. Vilanova, P. Balaguer and A. Ibeas, Systems Engineering and Automatic Control Group, 2008.
[16] Charles R. Nippert, Widener University, Simple Model that Illustrate DMC, 2002
47
APPENDICES
APPENDIX 1
MATLAB Coding
I. Input Variation Calculation
function [Iv]=inputvariation(MV) dif=filter([1 -1],[1],MV); absdif=abs(dif); Iv=sum(absdif);
II. Output Variation Calculation function IE=outputvariation(t,y,SP) %This function calculates the integral error IT1=trapz(t,y); IT2=trapz(t,SP); IE=IT2-IT1; III. Error Calculation
function IE=outputvariation(t,y,SP) %This function calculates the integral error IT1=trapz(t,y); IT2=trapz(t,SP); IE=IT2-IT1;
IV. Error Calculation for Model Predictive Controller
h = findobj(gcf, 'type', 'line'); xvalues = get(h, 'xdata'); yvalues = get(h, 'ydata'); p=cell2mat(yvalues(3));p=p'; q=cell2mat(xvalues(3));q=q'; ysp=1 e=abs(p-ysp); IAE=trapz(q,e) e2=e.^2; ISE=trapz(q,e2) e3=e.*q; ITAE=trapz(q,e3)
V. Transfer Function Ƭi = 2
>> Gp1=tf([-2 1],[4 1])
Gp2=tf([1],[10 1])
Transfer function:
-2 s + 1
--------
4 s + 1
Transfer function:
1
--------
10 s + 1
>> Gp=Gp1*Gp2
Transfer function:
-2 s + 1
-----------------
40 s^2 + 14 s + 1
>>
VI. Transfer Function Ƭi = 4
>> Gp1=tf([-4 1],[4 1])
Gp2=tf([1],[10 1])
Transfer function:
-4 s + 1
--------
4 s + 1
Transfer function:
1
--------
10 s + 1
>> Gp=Gp1*Gp2
Transfer function:
-4 s + 1
-----------------
40 s^2 + 14 s + 1
>>
VII. Transfer Function Ƭi = 6
>> Gp1=tf([-6 1],[4 1])
Gp2=tf([1],[10 1])
Transfer function:
-6 s + 1
--------
4 s + 1
Transfer function:
1
--------
10 s + 1
>> Gp=Gp1*Gp2
Transfer function:
-6 s + 1
-----------------
40 s^2 + 14 s + 1
>>
VIII. Transfer Function Ƭi = 8
>> Gp1=tf([-8 1],[4 1])
Gp2=tf([1],[10 1])
Transfer function:
-8 s + 1
--------
4 s + 1
Transfer function:
1
--------
10 s + 1
>> Gp=Gp1*Gp2
Transfer function:
-8 s + 1
-----------------
40 s^2 + 14 s + 1
>>